Question
What are the effects on the graph of the function f(x)=1x
when it is replaced by g(x)=−14x
?
Option #1: It reflects across the x-axis and vertically stretches by a factor of 4.
Option #2: It reflects across the x-axis and horizontally stretches by a factor of 4.
Option #3: It reflects across the x-axis and vertically compresses by a factor of 14
.
when it is replaced by g(x)=−14x
?
Option #1: It reflects across the x-axis and vertically stretches by a factor of 4.
Option #2: It reflects across the x-axis and horizontally stretches by a factor of 4.
Option #3: It reflects across the x-axis and vertically compresses by a factor of 14
.
Answers
GPT-4o mini
To analyze the transformation from \( f(x) = x \) to \( g(x) = -14x \), let's break it down step by step.
1. **Reflecting across the x-axis**: The negative sign in front of \( 14x \) indicates that the graph will reflect across the x-axis. So, for any \( x \) value, the output of \( g(x) \) will be the negative of \( f(x) \).
2. **Vertical stretching**: The \( 14 \) in front of \( x \) indicates that the graph is not only stretched, but it is actually vertically stretched by a factor of \( 14 \). This means that for any given \( x \), the output of \( g(x) \) will be \( 14 \) times the output of the original function \( f(x) \) (before considering the reflection).
Combining these transformations:
- The negative sign causes a reflection across the x-axis.
- The presence of \( 14 \) causes a vertical stretch by a factor of \( 14 \).
Therefore, the correct option is **Option #1: It reflects across the x-axis and vertically stretches by a factor of 14.**
(Note: If the vertical stretch factor mentioned in Option #1 is actually supposed to be a factor of 14, but it mentions a factor of 4, then this option would technically be incorrect. As it stands, if there is no correction to the optional description, the accurate transformation you've described is indeed a reflection and a vertical stretch, but it's by a factor of 14.)
1. **Reflecting across the x-axis**: The negative sign in front of \( 14x \) indicates that the graph will reflect across the x-axis. So, for any \( x \) value, the output of \( g(x) \) will be the negative of \( f(x) \).
2. **Vertical stretching**: The \( 14 \) in front of \( x \) indicates that the graph is not only stretched, but it is actually vertically stretched by a factor of \( 14 \). This means that for any given \( x \), the output of \( g(x) \) will be \( 14 \) times the output of the original function \( f(x) \) (before considering the reflection).
Combining these transformations:
- The negative sign causes a reflection across the x-axis.
- The presence of \( 14 \) causes a vertical stretch by a factor of \( 14 \).
Therefore, the correct option is **Option #1: It reflects across the x-axis and vertically stretches by a factor of 14.**
(Note: If the vertical stretch factor mentioned in Option #1 is actually supposed to be a factor of 14, but it mentions a factor of 4, then this option would technically be incorrect. As it stands, if there is no correction to the optional description, the accurate transformation you've described is indeed a reflection and a vertical stretch, but it's by a factor of 14.)